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## G = C2×Q8⋊Dic3order 192 = 26·3

### Direct product of C2 and Q8⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C2×Q8⋊Dic3
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — Q8⋊Dic3 — C2×Q8⋊Dic3
 Lower central SL2(𝔽3) — C2×Q8⋊Dic3
 Upper central C1 — C23

Generators and relations for C2×Q8⋊Dic3
G = < a,b,c,d,e | a2=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=c, ebe-1=b2c, dcd-1=bc, ede-1=d-1 >

Subgroups: 323 in 97 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C8, C2×C4, Q8, Q8, C23, Dic3, C2×C6, C4⋊C4, C2×C8, C22×C4, C2×Q8, C2×Q8, C2×Q8, SL2(𝔽3), C2×Dic3, C22×C6, Q8⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C2×SL2(𝔽3), C2×SL2(𝔽3), C22×Dic3, C2×Q8⋊C4, Q8⋊Dic3, C22×SL2(𝔽3), C2×Q8⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C2×Dic3, S4, CSU2(𝔽3), GL2(𝔽3), A4⋊C4, C2×S4, Q8⋊Dic3, C2×CSU2(𝔽3), C2×GL2(𝔽3), C2×A4⋊C4, C2×Q8⋊Dic3

Smallest permutation representation of C2×Q8⋊Dic3
On 64 points
Generators in S64
(1 10)(2 9)(3 15)(4 16)(5 7)(6 8)(11 13)(12 14)(17 51)(18 52)(19 47)(20 48)(21 49)(22 50)(23 44)(24 45)(25 46)(26 41)(27 42)(28 43)(29 64)(30 59)(31 60)(32 61)(33 62)(34 63)(35 57)(36 58)(37 53)(38 54)(39 55)(40 56)
(1 18 11 24)(2 21 12 27)(3 36 8 33)(4 39 7 30)(5 59 16 55)(6 62 15 58)(9 49 14 42)(10 52 13 45)(17 25 23 19)(20 28 26 22)(29 40 38 31)(32 37 35 34)(41 50 48 43)(44 47 51 46)(53 57 63 61)(54 60 64 56)
(1 20 11 26)(2 17 12 23)(3 38 8 29)(4 35 7 32)(5 61 16 57)(6 64 15 54)(9 51 14 44)(10 48 13 41)(18 22 24 28)(19 27 25 21)(30 34 39 37)(31 36 40 33)(42 46 49 47)(43 52 50 45)(53 59 63 55)(56 62 60 58)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20 21 22)(23 24 25 26 27 28)(29 30 31 32 33 34)(35 36 37 38 39 40)(41 42 43 44 45 46)(47 48 49 50 51 52)(53 54 55 56 57 58)(59 60 61 62 63 64)
(1 7 2 8)(3 11 4 12)(5 9 6 10)(13 16 14 15)(17 36 20 39)(18 35 21 38)(19 40 22 37)(23 33 26 30)(24 32 27 29)(25 31 28 34)(41 59 44 62)(42 64 45 61)(43 63 46 60)(47 56 50 53)(48 55 51 58)(49 54 52 57)

G:=sub<Sym(64)| (1,10)(2,9)(3,15)(4,16)(5,7)(6,8)(11,13)(12,14)(17,51)(18,52)(19,47)(20,48)(21,49)(22,50)(23,44)(24,45)(25,46)(26,41)(27,42)(28,43)(29,64)(30,59)(31,60)(32,61)(33,62)(34,63)(35,57)(36,58)(37,53)(38,54)(39,55)(40,56), (1,18,11,24)(2,21,12,27)(3,36,8,33)(4,39,7,30)(5,59,16,55)(6,62,15,58)(9,49,14,42)(10,52,13,45)(17,25,23,19)(20,28,26,22)(29,40,38,31)(32,37,35,34)(41,50,48,43)(44,47,51,46)(53,57,63,61)(54,60,64,56), (1,20,11,26)(2,17,12,23)(3,38,8,29)(4,35,7,32)(5,61,16,57)(6,64,15,54)(9,51,14,44)(10,48,13,41)(18,22,24,28)(19,27,25,21)(30,34,39,37)(31,36,40,33)(42,46,49,47)(43,52,50,45)(53,59,63,55)(56,62,60,58), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20,21,22)(23,24,25,26,27,28)(29,30,31,32,33,34)(35,36,37,38,39,40)(41,42,43,44,45,46)(47,48,49,50,51,52)(53,54,55,56,57,58)(59,60,61,62,63,64), (1,7,2,8)(3,11,4,12)(5,9,6,10)(13,16,14,15)(17,36,20,39)(18,35,21,38)(19,40,22,37)(23,33,26,30)(24,32,27,29)(25,31,28,34)(41,59,44,62)(42,64,45,61)(43,63,46,60)(47,56,50,53)(48,55,51,58)(49,54,52,57)>;

G:=Group( (1,10)(2,9)(3,15)(4,16)(5,7)(6,8)(11,13)(12,14)(17,51)(18,52)(19,47)(20,48)(21,49)(22,50)(23,44)(24,45)(25,46)(26,41)(27,42)(28,43)(29,64)(30,59)(31,60)(32,61)(33,62)(34,63)(35,57)(36,58)(37,53)(38,54)(39,55)(40,56), (1,18,11,24)(2,21,12,27)(3,36,8,33)(4,39,7,30)(5,59,16,55)(6,62,15,58)(9,49,14,42)(10,52,13,45)(17,25,23,19)(20,28,26,22)(29,40,38,31)(32,37,35,34)(41,50,48,43)(44,47,51,46)(53,57,63,61)(54,60,64,56), (1,20,11,26)(2,17,12,23)(3,38,8,29)(4,35,7,32)(5,61,16,57)(6,64,15,54)(9,51,14,44)(10,48,13,41)(18,22,24,28)(19,27,25,21)(30,34,39,37)(31,36,40,33)(42,46,49,47)(43,52,50,45)(53,59,63,55)(56,62,60,58), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20,21,22)(23,24,25,26,27,28)(29,30,31,32,33,34)(35,36,37,38,39,40)(41,42,43,44,45,46)(47,48,49,50,51,52)(53,54,55,56,57,58)(59,60,61,62,63,64), (1,7,2,8)(3,11,4,12)(5,9,6,10)(13,16,14,15)(17,36,20,39)(18,35,21,38)(19,40,22,37)(23,33,26,30)(24,32,27,29)(25,31,28,34)(41,59,44,62)(42,64,45,61)(43,63,46,60)(47,56,50,53)(48,55,51,58)(49,54,52,57) );

G=PermutationGroup([[(1,10),(2,9),(3,15),(4,16),(5,7),(6,8),(11,13),(12,14),(17,51),(18,52),(19,47),(20,48),(21,49),(22,50),(23,44),(24,45),(25,46),(26,41),(27,42),(28,43),(29,64),(30,59),(31,60),(32,61),(33,62),(34,63),(35,57),(36,58),(37,53),(38,54),(39,55),(40,56)], [(1,18,11,24),(2,21,12,27),(3,36,8,33),(4,39,7,30),(5,59,16,55),(6,62,15,58),(9,49,14,42),(10,52,13,45),(17,25,23,19),(20,28,26,22),(29,40,38,31),(32,37,35,34),(41,50,48,43),(44,47,51,46),(53,57,63,61),(54,60,64,56)], [(1,20,11,26),(2,17,12,23),(3,38,8,29),(4,35,7,32),(5,61,16,57),(6,64,15,54),(9,51,14,44),(10,48,13,41),(18,22,24,28),(19,27,25,21),(30,34,39,37),(31,36,40,33),(42,46,49,47),(43,52,50,45),(53,59,63,55),(56,62,60,58)], [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18,19,20,21,22),(23,24,25,26,27,28),(29,30,31,32,33,34),(35,36,37,38,39,40),(41,42,43,44,45,46),(47,48,49,50,51,52),(53,54,55,56,57,58),(59,60,61,62,63,64)], [(1,7,2,8),(3,11,4,12),(5,9,6,10),(13,16,14,15),(17,36,20,39),(18,35,21,38),(19,40,22,37),(23,33,26,30),(24,32,27,29),(25,31,28,34),(41,59,44,62),(42,64,45,61),(43,63,46,60),(47,56,50,53),(48,55,51,58),(49,54,52,57)]])

32 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6G 8A ··· 8H order 1 2 ··· 2 3 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 size 1 1 ··· 1 8 6 6 6 6 12 12 12 12 8 ··· 8 6 ··· 6

32 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 3 3 3 4 4 type + + + + - + - + + - + image C1 C2 C2 C4 S3 Dic3 D6 CSU2(𝔽3) GL2(𝔽3) S4 A4⋊C4 C2×S4 CSU2(𝔽3) GL2(𝔽3) kernel C2×Q8⋊Dic3 Q8⋊Dic3 C22×SL2(𝔽3) C2×SL2(𝔽3) C22×Q8 C2×Q8 C2×Q8 C22 C22 C23 C22 C22 C22 C22 # reps 1 2 1 4 1 2 1 4 4 2 4 2 2 2

Matrix representation of C2×Q8⋊Dic3 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 25 0 0 0 0 38 61
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 26 35 0 0 0 0 62 47
,
 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 1 72 0 0 0 0 1 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 27 0 0 0 0 27 0 0 0 0 0 0 0 68 59 0 0 0 0 54 5

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,38,0,0,0,0,25,61],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,62,0,0,0,0,35,47],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,27,0,0,0,0,27,0,0,0,0,0,0,0,68,54,0,0,0,0,59,5] >;

C2×Q8⋊Dic3 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes {\rm Dic}_3
% in TeX

G:=Group("C2xQ8:Dic3");
// GroupNames label

G:=SmallGroup(192,977);
// by ID

G=gap.SmallGroup(192,977);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,28,451,1684,655,172,1013,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^6=1,c^2=b^2,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*c*e^-1=b^-1,d*b*d^-1=c,e*b*e^-1=b^2*c,d*c*d^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

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